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Bounds of discriminants of number fields

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Abstract

In this survey, we introduce some progress on bounds of discriminants of number fields which contains the Rogers-Mulholland’s improvement of Minkowski’s classic inequality obtained by using geometry of numbers, Stark’s analytic method, improvement of analytic method by Odlyzko, and our results respectively.

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References

  1. J. V. Armitage, “Zeta functions with a zero \(s = \tfrac{1} {2}\),” Invent. Math. 15, 199–205 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  2. K. Chandrasekharan and R. Narasimhan, “The approximate functional equation for a class of zetafunctions,” Math. Ann. 152, 30–64 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  3. P. C. Hu, “An application of the Carleman formula,” Hunan Annals Math. [in Chinese] 10(1–2), 152–156 (1990).

    MATH  Google Scholar 

  4. P. C. Hu, “The properties of zero point distribution for Riemann’s zeta-function,” Pure Appl. Math. [in Chinese] 6(2), 6–12 (1990).

    MATH  Google Scholar 

  5. P. C. Hu and C. C. Yang, Value Distribution Theory Related to Number Theory (Birkhäuser, 2006).

    MATH  Google Scholar 

  6. P. C. Hu and Z. Ye, “A bound for the discriminant of number fields,” Proc. Amer. Math. Soc. 139, 2007–2008 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  7. P. C. Hu and Z. Ye, “The generalized Riemann Hypothesis and the discriminant of number fields,” J. Math. Anal. Appl. (2012), doi:10.1016/j.jmaa.2012.02.060.

    Google Scholar 

  8. H. P. Mulholland, “On the product of n complex homogeneous linear forms,” J. London Math. Soc. 35, 241–250 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. M. Odlyzko, “Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results,” Séminaire de Théorie des Nombres, 1–15 (Bordeaux, 1989).

    Google Scholar 

  10. C. A. Rogers, “The product of n real homogeneous linear forms,” Acta Math. 82, 185–208 (1950).

    Article  MathSciNet  MATH  Google Scholar 

  11. H. M. Stark, “Some effective cases of the Brauer-Siegel theorem,” Invent. Math. 23, 135–152 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  12. H. M. Stark, “The analytic theory of algebraic numbers,” Bull. Am. Math. Soc. 81, 961–972 (1975).

    Article  MATH  Google Scholar 

  13. J. Steuding, Value-Distribution of L-functions, Lecture Notes in Mathematics 1877 (Springer, 2007).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shandong University, Jinan, 250100, Shandong, P. R. China

    Pei-Chu Hu

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  1. Pei-Chu Hu
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Correspondence to Pei-Chu Hu.

Additional information

Based on talk at the International Workshop on p-Adic Methods for Modelling of Complex Systems, Bielefeld, April 15–19, 2013.

The text was submitted by the author in English.

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Hu, PC. Bounds of discriminants of number fields. P-Adic Num Ultrametr Anal Appl 5, 302–312 (2013). https://doi.org/10.1134/S2070046613040043

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  • DOI: https://doi.org/10.1134/S2070046613040043

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